Interdisciplinary Applied Mathematics

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Reynolds Equation in the Transition Flow Regime

Gas lubrication effects had been a primary interest for scientists long before the advent of the MEMS technology. For example, the work of (Burgdorfer, 1959) was motivated by lubrication characteristics in the slip flow regime. Similarly, (Hsia and Domoto, 1983) performed a series of experiments for

0.04 < Kn < 2.51 using different gases in order to change the mean free path, with bearing gaps of 0.075 p,m < ho < 1.6 p,m. These authors also derived a second-order slip boundary condition given in Table 2.2, and compared experimental results with the predictions of the Reynolds equation employing their second-order slip model for a Winchester type slider mechanism. In the analysis of Hsia and Domoto, the second-order slip model based on    the    continuum    approximation    was    applied    at    the    limit    of    the

transition flow regime. Although a reasonable match between the numerical and    experimental    results    for    integral    quantities    like    the    bearing    load

capacity was obtained, deviations in the squeezed film damping characteristics between the slip and the transition flow regimes exist. For example, the streamwise momentum equation (6.4) is based on the continuum-based constitutive models of viscous stress tensor. This is the starting point of the (Navier-Stokes-based) Reynolds equation, and it should be modified in the transition flow regime, using the viscous stress tensor of the Burnett equations given in equation (2.25). An alternative approach is to utilize Grad’s 13-moment equations (Grad, 1949), as demonstrated in (Chan and Sun, 2003).

The squeezed film problem can be thought of as a combination of pressure-driven and shear-driven flows. Therefore, one can foresee the differences between the Navier-Stokes and Burnett level analyses by separately analyzing the pressure- and shear-driven flow conditions. In Section 4.2.1 we presented the asymptotic analysis of pressure-driven flows in large aspect ratio channels under isothermal conditions as a function of e = ho/L ^ 0.

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